In spite of your adamant conclusion, I would've liked this being tested. In a fashion of some Discovery Channel's older programs. The battle royale of marketing practices.

Having the design of this bag, the check out zone, the cart under control, one can dictate how it would work. You can even place it on the bottom and have it automatically scanned as it rolls through a checkout table. Or, if it stays in a cart (and you can adjust it to be 1:1 to cart size), a chip on the side can also be read by the scanner on a side of said table. So many variants.

I don't see how it's more confusing than having all these loyalty cards. If it's a general store like Walmart, Auchan you visit daily (or weekly) and buy most things here, it's not really a problem. If they are freely interchangeable and have just one tier of price-cutting (like my cards in a local Lenta general store do), you don't need bags for every person in your household, you can bring any one at hand.

And it should be build at expense of the store and have a symbolic price, say $1, but being alike to IKEA ones - those I pick with myself everywhere by choice because I know they would carry 30kg of groceries if I'd need that. This investment into loyalty of returning customers and having brand being associated with zero waste wouldn't be for nothing. If it's not a metropolitan area, but a rural one, I think these bags would be, just like Xerox, a new coloquial naming to all good bags one encounter. They'd carry your ad everywhere, and everyone would know it's from Your-Store. And having that thing at hand promising you better prices at Your-Store, you'd probably come here, not to Other-Store.